Connecting concepts to problem-solving
Stephen Kanim
Department of Physics, New Mexico State University, Las Cruces, New Mexico 88003
Traditional quantitative problems of
the type commonly found at the end of
chapters in physics textbooks are
assigned to students most introductory
physics courses. Many students use a
formula-driven approach to solve these
problems that does not rely on
understanding
underlying
physics
concepts and that does little to
encourage the problem-solving skills
employed by experts. In another paper
presented at this conference, we gave an
example from electric circuits to
illustrate the use of “bridging exercises”
as part of students’ homework to
encourage students to solve problems by
starting with developed physics
concepts and models.1 In this paper, we
describe our attempts to use the same
approach in the context of electrostatics.
Previous Research
Much of what we know about student
problem solving in physics is a result of
‘expert/novice’ studies, in which the
problem solving strategies of experts
(typically university physics faculty
members) are compared to those of
novices
(typically
students
in
introductory
physics
courses).2
Researchers have found important
differences:
 Experts arrange their knowledge in a
hierarchical fashion with more
general ideas related to progressively
more specific pieces of information.
Novices have less knowledge and
their knowledge tends to be sparsely
connected and poorly organized.
 Experts tend to classify physics
problems based on underlying
concepts. In contrast, novices tend to
classify problems based on the
surface features of the problem.
 Experts use multiple representations
to clarify a problem. Mathematical
solution
typically
follows
a
restatement of the problem based on
general principles.
Novices often
start with an equation, substituting
given values and then looking for
additional equations to give other
values.
Many students see the
concepts as offering at best some
guidelines for selection of equations.
Results reported from the Maryland
Physics Expectations Survey of students
enrolled in introductory physics courses
are also pertinent to understanding
student approaches to problem solving.3
After completing an introductory
mechanics course, only about 50% of
the students who completed an
introductory mechanics course at a
large public university agreed with the
statement “When I solve most exam or
homework problems, I explicitly think
about the concepts that underlie the
problem.” Many students do not seem
to make a strong connection between
the concepts and the problems that are
part of introductory physics courses.
Effect of conceptual development on
problem solving -- electric circuits
Does an increased emphasis on the
development of concepts in the
introductory course promote more
expert-like problem solving? In the
context of electric circuits, we found
that students often did not apply learned
concepts to quantitative problems. We
designed homework worksheets that
included "bridging exercises" intended
to form a more explicit link between
concepts developed in tutorials and endof chapter problems. After completing
the worksheets, more students were able
to solve quantitative problems related to
current in electric circuits.1
Bridging exercises for other topics
Encouraged by our experience with
electric circuits, we wrote worksheets
for other topics in the calculus-based
electricity and magnetism course. These
worksheets also attempted to guide
students in their consideration of the
concepts underlying more traditional
problems. For many of these topics,
however, comparatively little research
into specific student difficulties existed
upon which to base the worksheets. We
tried to anticipate student’s difficulties
with the concepts and with the
mathematical steps required to solve
commonly
assigned
quantitative
problems, and to encourage solution of
these problems without resorting to
formula-driven approaches.
We offered optional problem-solving
sessions for students in sections in
which the worksheets were first
assigned. Students worked through the
assignments in small groups, with
instructors
available
to
answer
questions. Our observations of these
small groups led us to realize that for
many topics the worksheets were poorly
matched to students' needs.
As an example, we discuss our
experience with a worksheet intended to
guide students through a standard
problem in electrostatics shown in
Figure 1. The steps required to solve
this problem include breaking the rod up
into small elements of length dx, finding
the electric field due to one of these
charge elements, calculating the
component of this field perpendicular to
the rod, and then integrating to find the
total electric field.
P
y
L
A thin nonconducting rod of length L
carries a total charge q, spread uniformly
along it. Show that E at point P on the
perpendicular bisector in the figure is
given by
q
1
E=
2 oy (L 2 + 4 y2)1/2
Figure 1. Electric field problem.
The initial steps of the homework
worksheet are shown in Figure 2. These
questions were intended to allow
students to reason qualitatively about
factors that determine the electric field.
Criticize the following statement:
"From Coulomb's law, we would expect a
force on a positive charge qt placed at
point P to have a magnitude
1 q1q2
1 q1qt
F=
.
=
2
4 o r
4 o y2
So the electric field at point P is the force
divided by the test charge:
1 q
F
."
E= q =
4 o y2
t
Would you expect the electric field at point
P to be larger or smaller than the value that
would be obtained using the method above?
Figure 2.
Initial questions
electrostatics worksheet.
for
We expected that these questions
would be relatively easy for students
(especially compared to the more
mathematical steps that would follow).
Most of the students in the optional
problem-solving session, however, were
unable to answer these questions even
after significant guidance. It became
clear that we did not have an adequate
understanding of student’s conceptual
difficulties in electrostatics to allow us
to
design
effective
“bridging”
worksheets. We decided that if we
hoped to improve students’ ability to
solve
quantitative
problems
in
electrostatics in a more expert-like
manner we would first need to
understand more about the nature of the
underlying conceptual difficulties.
Conceptual difficulties in electrostatics
We found that for electrostatics,
many difficulties exhibited by students
were difficulties with the addition of
vectors.4 Many students treated vectors
as scalar quantities when they were
asked to add them. Other students were
able to show graphically what the sum
of vectors would be, but then gave
algebraic answers that were inconsistent
with their drawings but were consistent
with treating the vectors as scalars. Still
other students added only the
component of a vector that was collinear
with the other vector in a vector sum,
neglecting the effect of perpendicular
components.
Additional difficulties
with the relationship between vectors
and their components were elicited by
some of the questions that we asked as
part of this study. Many of these
difficulties with vector addition were
also elicited in topics other than
electrostatics.
For example, after
noticing that some students claimed that
vectors of equal magnitude representing
electrostatic forces or fields completely
cancelled one another (in cases where
they were pointing at an acute angle to
one another), we asked a similar
question in the context of momentum in
a mechanics class. This same difficulty
was exhibited by about the same
proportion of students.
We also discovered difficulties that
were specific to electrostatics. For
example, many students answer
questions in a manner that is consistent
with a belief that electric fields can be
blocked by the presence of nearby
conductors or insulators. Some students
reason that insulators act to insulate a
charge from the effects of an electric
field.
Other students give similar
reasoning about the effect of conductors.
Some of the difficulties revealed by
our investigation were difficulties with
fundamental underlying concepts. For
example, many students had difficulty
differentiating between charge and
charge density. We asked students the
questions shown in Figure 3 after the
completion
of
instruction
in
electrostatics up to Gauss’ law.
A plastic b lock o f length
w, h eig ht h, and thickness
t contains a net pos itiv e h
charg e Qo d istributed
un iformly throug hout its
vo lume.
A. What is the vo lume charge
dens ity o f the plastic b lock?
The block is no w
bro ken in to two p ieces .
The pieces are labeled
A an d B as sho wn .
h
B. Ran k th e v olume
charg e d ens ities of the
2w
original b lock ( o),
3
piece A (A), and piece
B ( B). Explain your
reas onin g.
w
t
w
3
t
Figure 3. Charged block question.
About 70% were able to find the
charge density from the given
information. However, only about 55%
recognized that the charge densities of
the pieces would be the same. About
20% answered that piece B would have
the largest charge density.
Often
student's explanations suggested that
they were thinking in terms of the
equation for charge density, but were
holding the charge constant: Similar
difficulties with expressions involving
more than 2 variables have been noted
in different contexts, for example the
ideal gas law.
About 20% answered that the original
piece would have the largest charge
densities.
These responses often
included reasoning that suggested a
difficulty differentiating between charge
and charge density.
Student
difficulties
with
the
interpretation of information about
charge density were also noted when
students were asked to solve problems
involving distributed charges.
For
example, on examination questions
where students were expected to use
Gauss’ law, only about 30% were able
to find the enclosed charge. Many of
the incorrect responses included a
substitution of the given charge density
information into equations that required
a charge.
Summary
unanticipated difficulties, often with
fundamental underlying concepts. If we
hope to improve students' ability to
solve problems in a more expert-like
manner, we need to understand the
conceptual issues associated with the
problems we are posing, and we need to
have developed effective methods for
addressing these conceptual issues.
Otherwise, student difficulties with
problem solving will often reflect
conceptual errors with the underlying
material.
Acknowledgements
I would like to thank the many
students who participated in the studies
described in this paper. The assistance
of all members of the Physics Education
Group at the University of Washington
is gratefully acknowledged, and in
particular the substantial contributions
made by Peter Shaffer and Lillian
McDermott.
References
1
Where
common
conceptual
difficulties for a certain topic in physics
are well understood, it is possible to
design curricula that addresses many of
these difficulties. We have found that in
these cases an explicit link between the
developed concepts and associated
quantitative
problems
(“bridging
exercises”) increased the number of
students who were able to solve
quantitative problems. Moreover, the
approach taken by students in these
cases was often less formula-driven and
was more likely to include explicit
reference to developed concepts.
In electricity and magnetism,
however, we found that often we did not
have a sufficient understanding of
student’s conceptual difficulties to
design effective bridging exercises. For
many topics, student's exhibited
S. Kanim, "Connecting concepts
about current to quantitative circuit
problems," in these conference
proceedings.
2
J. Larkin, J. McDermott, D. Simon, and
H. Simon, “Expert and novice
performance
in
solving
physics
problems,” Science 208(20), 1335-1342
(1980).
3
E. Redish, J. Saul, and R. Steinberg,
“Student expectations in introductory
physics,” American Journal of Physics
66(3), 212-224 (1998).
4
S. Kanim, “An investigation of
student difficulties in qualitative and
quantitative
problem
solving:
Examples from electric circuits and
electrostatics,” Ph.D. dissertation,
Department of Physics, University
of Washington (1999).